Factors, Multiples and Primes - Short Problems

A collection of short Stage 3 and 4 problems on factors, multiples and primes



Four or Five

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 19 - 2012
The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?

Seven from Nine

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 21 - 2013
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

Factor Trio

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 45 - 2013
Which of the numbers shown is the product of exactly 3 distinct prime factors?

Back of the Queue

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?

Calculation 2000

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

Weekly Problem 10 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

Weekly Problem 14 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?

Weekly Problem 22 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

The following sequence continues indefinitely... Which of these integers is a multiple of 81?

Weekly Problem 26 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

Weekly Problem 41 - 2008

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

How many pairs of numbers of the form x, 2x+1 are there in which both numbers are prime numbers less than 100?

School Dinners

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 1 - 2009
Our school dinners offer the same choice each day, and each day I try a new option. How long will it be before I eat the same meal again?

Little Goldbach

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 11 - 2009
How many of the numbers 1 to 20 are not the sum of two primes?

Trailing Zeros

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 13 - 2009
How many zeros does 50! have at the end?

Weekly Problem 41 - 2009

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

At a cinema a child's ticket costs £$4.20$ and an adult's ticket costs £$7.70$. How much did is cost this group of adults and children to see a film?

Weekly Problem 45 - 2009

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

Three Primes

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

Square LCM

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 7 - 2010
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Factorised Factorial

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

17s and 23s

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 22 - 2010
Can you form this 2010-digit number...

HCF Expression

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 30 - 2010
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Product 100

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 38 - 2010
The product of four different positive integers is 100. What is the sum of these four integers?

Factor List

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 48 - 2010
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Adjacent Factors

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 5 - 2011
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Sticky Fingers

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 7 - 2011
Roo wants to puts stickers on the cuboid he has made from little cubes. Will he have any stickers left over?

Square Product

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 10 - 2011
Will this product give a perfect square?

Indivisible

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 25 - 2011
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

Many Clues, One Answer

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 35 - 2011
You are given lots of clues about a number. Can you work out what it is?

Divisible Palindrome

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 5 - 2014
What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

Digital Division

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 15 - 2014
How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?

Almost a Million

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 26 - 2014
Which of the given numbers are divisible by 6?

Supercomputer

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 28 - 2014
What is the units digit of the given expression?

One Less Remainder

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 39 - 2014
A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?

Primes and Six

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 1 - 2015
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.

Factor Sum

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 4 - 2015
Given any positive integer n, Paul adds together the factors of n, apart from n itself. Which of the numbers 1, 3, 5, 7 and 9 can never be Paul's answer?

Tricky Customer

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 6 - 2015
Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?

Added Power

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 33 - 2015
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?

Last-but-one

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 1 - 2016
What is the last-but-one digit of 99! ?

Birthday Tables

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 2 - 2016
How many tables of each type does Mark need at his birthday party?

Simon's Age

Short Challenge Level: Challenge Level:1

Weekly Problem 3 - 2016
How many times has Simon's age changed from a square to a prime?

Grandma's Cake

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 7 - 2016
What is the smallest number of pieces grandma should cut her cake into to guarentee each grandchild gets the same amount of cake and none is left over.

Divisible by Its Digit Sum

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 18 - 2016
The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?

End of a Prime

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 25 - 2016
A list is made of every digit that is the units digit of at least one prime number. How many digits appear in the list?

Stamp Collecting

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 28 - 2016
Last week, Tom and Sophie bought some stamps for their collections. Each stamp Tom bought cost him £1.10, whilst Sophie paid 70p for each of her stamps. Between them they spent exactly £10. How many stamps did they buy in total?

Threes and Fours

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 32 - 2016
What is the smallest integer which has every digit a 3 or a 4 and is divisible by both 3 and 4?

Square Percentage

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 36 - 2016
What percentage of the integers between 1 and 10,000 are square numbers?

354972

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 42 - 2016
What is the remainder when 354972 is divided by 7?

Knockdown

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 51 - 2016
Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?

Multiplication Table Puzzle

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 6 - 2017
In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?

Suit Sequence

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 8 - 2017
A pattern repeats every six symbols. What are the 100th and 101st symbols?

Peter's Primes

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 22 - 2017
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

Triangular Algebra

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 26 - 2017
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?

Primes 6 Apart

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 30 - 2017
The prime numbers p and q are the smallest primes that differ by 6. What is the value of p+q?

Smallest Abundant Number

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 34 - 2017
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?

Ones, Twos and Threes

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 5 - 2017
Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

Square Total

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 13 - 2017
Anastasia thinks of a number. Each of her friends performs an operation on it. The total of these is a square number. What is the smallest number Anastasia could have thought of?

What's on the Back?

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 21 - 2017
Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?

Common Remainder

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 35 - 2017
144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?

Long List

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?